In triangle ABC, M is the midpoint of line AB. Let D be the point on line BC such that line AD bisects angle BAC, and let the perpendicular bisector of line AB intersect line AD at P. If line AB = 36 and line MP = 9, then find the distance from P to line line AC.

The line PE representing the shortest distance from P to the line AC at E will be perpendicular to AC (Shortest distance from a point to a line = radius of a circle with center at the point and the line as tangent to the circle).

Intersect

When two or more lines cross each other in a plane, they are called intersecting lines. The intersecting lines share a common point, which exists on all the intersecting lines, and is called the point of intersection.

Triangle

A triangle is a polygon with three sides having three vertices. The angle formed inside the triangle is equal to 180 degrees. It means that the sum of the interior angles of a triangle is equal to 180°

Then

∠PEA = 90°

In ∠MAP = ∠EAP (angle bisected by AD)

∠AMP = ∠PEA = 90° (Angle between perpendicular lines)

In ΔPAM, ∠MAP + ∠AMP + ∠APM = 180°

Similarly in ΔPAE, ∠EAP + ∠PEA + ∠APE = 180°

∠MAP + ∠AMP = ∠EAP + ∠PEA, we have;

∠APM = ∠APE

ΔPAM ≅ ΔPAE ASA (congruent condition)

Side AP ≅ AP (reflective property)

Then

For right triangles ΔPAM and ΔPAE to be equivalent, MP = PE = 12.

MP = PE = 12

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